As mentioned above, I undertook a large translation project in collaboration with a colleague at another institution who is a mathematician and a native speaker of Arabic. We have been working on translating one of the most important mathematics books of the medieval Islamic civilization, *Miftāh **al-His**āb**,* which can be translated as either *Key to Arithmetic* or *Calculator’s Key,* by Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (1380-1429), who is known as al-Kāshī or al-Kāshānī. Researchers who have studied *Miftāh **al-His**āb**,* completed in 1427, agree that it is “the crowning achievement of Islamic arithmetic” ([3], p. 22). It has three main parts: one on arithmetic, one on geometry and measurement, and the third part on algebra. Among its many jewels are algorithms to find (approximate as rational numbers) square roots and higher order roots of integers.

**Figure 6.** Written in 1427, Al-Kāshī's *Miftāh al-Hisāb* is one of the most important mathematics books of the medieval Islamic civilization. These pages are from one of the oldest known manuscripts of *Miftāh al-Hisāb*, from Süleymaniye Library in Istanbul, written in 1450 (854 h.) (*Source:* Image used by kind permission of Süleymaniye Library, Istanbul, Turkey)

This book came to the attention of western researchers in 1948 ([15]). Prior to this date, Simon Stevin (1548-1620) was widely assumed to be the first inventor of the decimal fractions. However, *Miftāh **al-His**āb* contains a systematic treatment of the decimal fractions. Still, it was not al-Kāshī who first introduced decimal fractions. According to the best available research and original sources, decimal fractions were first introduced in the middle of the tenth century, almost five centuries before Al-Kāshī and over six centuries before Stevin, by the scholar Abu'l Hasan Ahmad ibn Ibrahim al-Uqlīdisī (920-980), known as Al-Uqlīdisī, in his book *Kitab al-fusul fi al-Hisāb al-Hindi (The Book of Chapters in Indian Arithmetic), *which has been translated into English [21].

The story of modern researchers’ understanding of the origin of decimal fractions is highly instructive, showing that the lack of research and lack of evidence from original sources can lead to an incorrect understanding of the history of mathematics. Furthermore, once misconceptions are widely circulated, it is then very difficult to correct them, as highly reputable people may repeat inaccurate information. As noted above, it takes more than scholarly articles in research journals to change the common narrative.

While our translation project is not yet complete, we noticed that the content of *Miftāh **al-His**āb* contains another potential connection between al-Kāshī and Stevin. We observed that Stevin also presented algorithms to find (again, to approximate) roots of integers. On the surface, the procedures by al-Kāshī and Stevin look very different. A close examination of their work, however, reveals that they actually used the same underlying algorithm. The details are described in our recent paper [1]. We also observed that Stevin’s procedure contains a flaw and does not always work. Moreover, he did not seem to understand the rationale behind at least one step in his algorithm, hence did not seem to have a full understanding of the algorithm he presented. So far, to the best of our knowledge, nobody has established a link between the two scholars, but, based on these observations, we wonder if there was any direct or indirect connection between Al-Kāshī and Stevin. Like many other questions in the history of science (such as if and how Copernicus came to know of the works of earlier Islamic scholars such as ibn Shātir and al-Tūsī [20], [7]), this is an open question at the moment.

An alternative method of approximating square roots of integers by a 12th century Islamic scholar, Abu Bakr al-Hassar, with the flexibility of improving the accuracy of approximation by increasing the number of iterations, is presented in the *Convergence* article, "Extracting Square Roots Made Easy: A Little Known Medieval Method" [9].